What Is the Mode?
In statistics, the mode is the value that appears most frequently in a dataset. It is the third of the three measures of central tendency, alongside the mean (average) and the median (middle value). While the mean balances all values equally and the median marks the midpoint, the mode tells you which value is the most common or most typical in the data — the one that occurs more than any other.
For ungrouped (raw) data, finding the mode is straightforward: simply look for the value that repeats the most. For example, in the set 2, 4, 5, 5, 6, 7, the value 5 appears twice while every other value appears only once — so the mode is 5.
Unimodal — one mode
Bimodal — two modes
Trimodal — three modes
Multimodal — four or more
No mode — all equal frequency
📌 Bimodal & Multimodal data: A dataset can have more than one mode. The set 2, 2, 2, 3, 4, 4, 5, 5, 5 has two modes — 2 and 5 — because both appear exactly three times. This is called bimodal data. Three modes = trimodal; four or more modes = multimodal. If every value appears the same number of times, there is no mode at all.
Mode of Grouped Data — The Formula
When data is grouped into class intervals, you cannot identify the mode directly from the table — individual values are hidden inside classes. Instead, you first locate the class with the highest frequency, called the modal class. The mode is then estimated using a formula that takes into account not just the modal class itself, but also how its frequency compares with the class just before it (f₀) and the class just after it (f₂).
Mode = l + [ (f₁ − f₀) / (2f₁ − f₀ − f₂) ] × h
where:
l = lower boundary of the modal class
f₁ = frequency of the modal class (the highest frequency)
f₀ = frequency of the class before the modal class (preceding class)
f₂ = frequency of the class after the modal class (succeeding class)
h = class width (size of each class interval)
💡 Why f₀ and f₂ matter: The formula does not simply place the mode at the midpoint of the modal class. Instead, the fraction (f₁ − f₀)/(2f₁ − f₀ − f₂) shifts the mode towards the lower end of the class if the class before it has a high frequency, and towards the upper end if the class after it has a high frequency. This makes the estimate more accurate — it reflects the actual "pull" of neighbouring classes on where the true mode lies within the modal class.
Textbook Worked Example — Family Size in 20 Households
Before the exercise questions, the textbook demonstrates the formula with a survey of 20 households and their family sizes. This example shows the complete method step by step.
| Family Size | 1 – 3 | 3 – 5 | 5 – 7 | 7 – 9 | 9 – 11 |
|---|
| No. of Families |
7 | 8 ← modal class | 2 | 2 | 1 |
Modal class:3 – 5
(highest frequency = 8)
| l | = 3 (lower boundary of modal class 3–5) |
| f₁ | = 8 (frequency of modal class) |
| f₀ | = 7 (frequency of preceding class 1–3) |
| f₂ | = 2 (frequency of succeeding class 5–7) |
| h | = 3 − 1 = 2 (class width) |
Mode = 3 + [(8 − 7) / (2×8 − 7 − 2)] × 2
= 3 + [1 / (16 − 9)] × 2
= 3 + (1/7) × 2
= 3 + 2/7
= 3.286
✅ Mode of family size data = 3.286
📌 Interpretation: The modal family size is approximately 3.3 members. This means the most commonly occurring household size in this locality is in the 3–5 family-member range, estimated to be around 3.3 — very close to the lower boundary because the preceding class (7 families) is nearly as large as the modal class (8 families), pulling the mode downward.
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The ages of patients admitted to a hospital on a particular day are distributed below. Find the mode and the mean of the data. Compare and interpret the two measures.
| Age (years) | 5–15 | 15–25 | 25–35 | 35–45 | 45–55 | 55–65 |
|---|
| No. of Patients |
6 | 11 | 21 | 23 ← modal | 14 | 5 |
Part A — Finding the Mode
Modal class:35 – 45
(highest frequency = 23)
| l | = 35 |
| f₁ | = 23 |
| f₀ | = 21 (class 25–35) |
| f₂ | = 14 (class 45–55) |
| h | = 10 |
Mode = 35 + [(23 − 21) / (2×23 − 21 − 14)] × 10
= 35 + [2 / (46 − 35)] × 10
= 35 + (2/11) × 10
= 35 + 1.818...
≈ 36.82 years
Part B — Finding the Mean (Direct Method)
| Age (C.I.) | fᵢ | xᵢ | fᵢ × xᵢ |
|---|
| 5 – 15 | 6 | 10 | 60 |
| 15 – 25 | 11 | 20 | 220 |
| 25 – 35 | 21 | 30 | 630 |
| 35 – 45 | 23 | 40 | 920 |
| 45 – 55 | 14 | 50 | 700 |
| 55 – 65 | 5 | 60 | 300 |
| Total | Σfᵢ = 80 | — | Σfᵢxᵢ = 2830 |
Mean = 2830 ÷ 80
= 35.375 ≈ 35.37 years
📊 Interpretation — Comparing Mode and Mean
The mode (36.82 years) tells us the single most common age among admitted patients — the age group 35–45 had the highest number of admissions. The mean (35.37 years) gives the overall average age across all 80 patients. Both values are close to each other and both fall in the 35–45 age bracket, indicating that middle-aged adults between 35 and 45 years are the most frequently hospitalised group in this locality.
The observed lifetimes (in hours) of 225 electrical components are distributed below. Determine the modal lifetime of the components.
| Lifetime (hrs) | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 |
|---|
| Frequency |
10 | 35 | 52 | 61 ← modal | 38 | 29 |
Modal class:60 – 80 hours
(highest frequency = 61)
| l | = 60 |
| f₁ | = 61 |
| f₀ | = 52 (class 40–60) |
| f₂ | = 38 (class 80–100) |
| h | = 20 |
Mode = 60 + [(61 − 52) / (2×61 − 52 − 38)] × 20
= 60 + [9 / (122 − 90)] × 20
= 60 + (9/32) × 20
= 60 + 180/32
= 60 + 5.625
= 65.625 hours
✅ Modal lifetime of electrical components = 65.625 hours
📌 Real-world significance: A modal lifetime of 65.625 hours means the most commonly observed failure point for these components is around 65–66 hours of operation. An engineer reviewing this data would flag this as the typical lifespan and design maintenance schedules accordingly — rather than using the mean, which can be skewed by a few very long-lasting or very short-lived components.
Monthly household expenditure data for 200 families of Gummadidala village is given. Find the modal monthly expenditure and the mean monthly expenditure.
| Expenditure (Rs.) | f |
|---|
| 1000 – 1500 | 24 |
| 1500 – 2000 | 40 ← modal class |
| 2000 – 2500 | 33 |
| 2500 – 3000 | 28 |
| 3000 – 3500 | 30 |
| 3500 – 4000 | 22 |
| 4000 – 4500 | 16 |
| 4500 – 5000 | 7 |
Part A — Finding the Mode
Modal class:1500 – 2000
(highest frequency = 40)
| l | = 1500 |
| f₁ | = 40 |
| f₀ | = 24 (class 1000–1500) |
| f₂ | = 33 (class 2000–2500) |
| h | = 500 |
Mode = 1500 + [(40 − 24) / (2×40 − 24 − 33)] × 500
= 1500 + [16 / (80 − 57)] × 500
= 1500 + (16/23) × 500
= 1500 + 347.83
= Rs. 1847.83
Part B — Finding the Mean (Step Deviation Method, h = 500, a = 2750)
| Expenditure (Rs.) | fᵢ | xᵢ | uᵢ = (xᵢ−2750)/500 | fᵢuᵢ |
|---|
| 1000 – 1500 | 24 | 1250 | −3 | −72 |
| 1500 – 2000 | 40 | 1750 | −2 | −80 |
| 2000 – 2500 | 33 | 2250 | −1 | −33 |
| 2500 – 3000 | 28 | 2750 | 0 (a) | 0 |
| 3000 – 3500 | 30 | 3250 | +1 | +30 |
| 3500 – 4000 | 22 | 3750 | +2 | +44 |
| 4000 – 4500 | 16 | 4250 | +3 | +48 |
| 4500 – 5000 | 7 | 4750 | +4 | +28 |
| Total | Σfᵢ = 200 | — | — | Σfᵢuᵢ = −35 |
Mean = 2750 + (−35/200) × 500
= 2750 + (−35 × 2.5)
= 2750 − 87.5
= Rs. 2662.50
📊 Interpretation
The mode (Rs. 1847.83) shows the most commonly occurring monthly household expenditure level — the majority of families spent around Rs. 1,848 per month. The mean (Rs. 2662.50), however, is significantly higher. This gap occurs because a smaller number of families with much higher expenditures (Rs. 3,000–5,000) pull the average upward, while the mode remains anchored to where the bulk of families actually spend. In such income/expenditure distributions, the mode is a better representative of the "typical" family's spending than the mean.
The state-wise teacher-student ratio in higher secondary schools of India is distributed below. Find the mode and mean of this data. Interpret the two measures.
| Students per Teacher | 15–20 | 20–25 | 25–30 | 30–35 | 35–40 | 40–45 | 45–50 | 50–55 |
|---|
| No. of States |
3 | 8 | 9 | 10 ← modal | 3 | 0 | 0 | 2 |
Part A — Finding the Mode
Modal class:30 – 35
(highest frequency = 10)
| l | = 30 |
| f₁ | = 10 |
| f₀ | = 9 (class 25–30) |
| f₂ | = 3 (class 35–40) |
| h | = 5 |
Mode = 30 + [(10 − 9) / (2×10 − 9 − 3)] × 5
= 30 + [1 / (20 − 12)] × 5
= 30 + (1/8) × 5
= 30 + 0.625
= 30.625
Part B — Finding the Mean (Assumed Mean Method, a = 32.5)
| Students/Teacher | fᵢ | xᵢ | dᵢ = xᵢ − 32.5 | fᵢdᵢ |
|---|
| 15 – 20 | 3 | 17.5 | −15 | −45 |
| 20 – 25 | 8 | 22.5 | −10 | −80 |
| 25 – 30 | 9 | 27.5 | −5 | −45 |
| 30 – 35 | 10 | 32.5 | 0 (a) | 0 |
| 35 – 40 | 3 | 37.5 | +5 | +15 |
| 40 – 45 | 0 | 42.5 | +10 | 0 |
| 45 – 50 | 0 | 47.5 | +15 | 0 |
| 50 – 55 | 2 | 52.5 | +20 | +40 |
| Total | Σfᵢ = 35 | — | — | Σfᵢdᵢ = −115 |
Mean = 32.5 + (−115 / 35)
= 32.5 − 3.286
≈ 29.22
📊 Interpretation
The mode (30.625 students per teacher) indicates the most common teacher-student ratio across Indian states — the majority of states have roughly 30–31 students per teacher. The mean (29.22) is slightly lower, pulled down by the two states with unusually high ratios (50–55) having only a minor effect, while the several states with lower ratios (15–30) bring the average down. Both measures suggest that the typical state in India has around 29–31 students per teacher in higher secondary schools.
⚠️ Zero-frequency classes: This question contains two classes (40–45 and 45–50) with frequency 0. For the mean calculation, these simply contribute 0 to Σfᵢdᵢ — but they still count as classes in the distribution. Do not skip them from the table. Note also that the last class (50–55, f = 2) acts as an outlier group and pulls the mean upward compared to the mode.
The distribution below shows the runs scored in ODI cricket by some of the world's top batsmen. Find the mode of the data.
| Runs |
3000–4000 | 4000–5000 | 5000–6000 | 6000–7000 |
7000–8000 | 8000–9000 | 9000–10000 | 10000–11000 |
| Batsmen |
4 | 18 ← modal | 9 | 7 | 6 | 3 | 1 | 1 |
Modal class:4000 – 5000 runs
(highest frequency = 18)
| l | = 4000 |
| f₁ | = 18 |
| f₀ | = 4 (class 3000–4000) |
| f₂ | = 9 (class 5000–6000) |
| h | = 1000 |
Mode = 4000 + [(18 − 4) / (2×18 − 4 − 9)] × 1000
= 4000 + [14 / (36 − 13)] × 1000
= 4000 + (14/23) × 1000
= 4000 + 608.695...
≈ 4608.7 runs
✅ Modal runs scored by top ODI batsmen = 4608.7 runs
💡 What this means: The most common career ODI run tally among these top batsmen falls around 4,609 runs — within the 4,000–5,000 range. The very large difference between f₁ (18) and f₀ (4) means the modal class stands out sharply above the preceding class, so the mode is shifted well into the class interval rather than sitting near the lower boundary.
A student counted cars passing a spot on a road for 100 three-minute periods. The results are summarised below. Find the mode of the data.
| Cars |
0–10 | 10–20 | 20–30 | 30–40 |
40–50 | 50–60 | 60–70 | 70–80 |
| Frequency |
7 | 14 | 13 | 12 |
20 ← modal | 11 | 15 | 8 |
Modal class:40 – 50 cars
(highest frequency = 20)
| l | = 40 |
| f₁ | = 20 |
| f₀ | = 12 (class 30–40) |
| f₂ | = 11 (class 50–60) |
| h | = 10 |
Mode = 40 + [(20 − 12) / (2×20 − 12 − 11)] × 10
= 40 + [8 / (40 − 23)] × 10
= 40 + (8/17) × 10
= 40 + 4.706...
≈ 44.7 cars
✅ Mode of cars passing through in a 3-minute period = 44.7 cars
📌 Traffic interpretation: The modal number of cars passing this spot in any 3-minute window is approximately 45 cars. A traffic engineer would use this value — not the mean — to design signal timing, because the mode tells them the most typical flow rate rather than the mathematical average (which could be distorted by a few unusually busy or quiet periods).
Quick Summary — All 6 Questions at a Glance
| Q |
Dataset |
Modal Class |
Mode |
Mean (if asked) |
| 1 | Hospital patient ages (n=80) | 35 – 45 | 36.82 years | 35.37 years |
| 2 | Component lifetimes (n=225) | 60 – 80 hrs | 65.625 hrs | — |
| 3 | Household expenditure (n=200) | 1500 – 2000 | Rs. 1847.83 | Rs. 2662.50 |
| 4 | Teacher-student ratio (n=35) | 30 – 35 | 30.625 | 29.22 |
| 5 | ODI cricket runs (top batsmen) | 4000 – 5000 | 4608.7 runs | — |
| 6 | Cars passing a spot (n=100) | 40 – 50 | 44.7 cars | — |
Common Mistakes to Avoid
- Picking f₀ and f₂ incorrectly: f₀ is always the class immediately before the modal class, and f₂ is immediately after. A common error is picking the class with the second-highest frequency instead. Position matters, not rank.
- Using the wrong lower boundary (l): The lower boundary l is the lower limit of the modal class, not of the entire dataset. Always read l directly from the modal class row in your table.
- Confusing class width h with class count: h is the numerical width of each class (e.g. 80 − 60 = 20 for the 60–80 class), not how many classes there are. If classes have equal width, h is the same for all; use that single value.
- Getting a negative denominator: 2f₁ − f₀ − f₂ should always be positive (since f₁ is the highest frequency). If you compute a negative or zero denominator, you have made an arithmetic error — recheck your subtraction.
- Identifying the wrong modal class when two frequencies are equal: If two classes share the highest frequency, the data is bimodal (two modes). Do not arbitrarily pick one — the question or context will specify which to use, or you may need to compute both.
- Using midpoint instead of lower boundary for l: The formula requires the lower boundary of the modal class, not its midpoint (class mark). These are different values — confusing them gives a completely wrong answer.
⛔ Board exam alert (CBSE, Telangana & AP): Mode questions almost always appear as a combined Mode + Mean or Mode + Median question worth 4–5 marks. The most commonly tested pattern is Question 1's style — compute both mode and mean, then write a two-sentence interpretation comparing them. Practice writing the interpretation in your own words, since examiners specifically award 1 mark for a correct, contextual comparison.
What This Exercise Prepares You For
Having mastered the mode formula for grouped data, you are ready to complete your understanding of all three measures of central tendency. The related lessons in this chapter are the Statistics Introduction (which covers the mean methods), Exercise 14.1 (Mean), and Exercise 14.3 (Median), where cumulative frequency tables and the ogive graph are used. Together, the mean, median, and mode give a complete picture of any dataset.
The mode concept you have practised here also underpins discussions in Probability — the most likely outcome in a probability experiment corresponds to the modal value in the distribution of outcomes.
📐 Board Exam Tip: The mode formula has five variables — l, f₁, f₀, f₂, and h. In your working, always identify and state all five values explicitly before substituting into the formula. Examiners award 1 mark for correctly identifying the modal class and 1 mark for correctly listing all five values, even if your final arithmetic has a small slip.
